# How do you differentiate f(x)= e^x/(e^(3x) -5x) using the quotient rule?

Jun 23, 2016

$f ' \left(x\right) = \frac{{e}^{x} \left(5 - 5 x - 2 {e}^{3 x}\right)}{{e}^{3 x} - 5 x} ^ 2.$

#### Explanation:

The Quotient Rule : $\left(\frac{u}{v}\right) ' = \frac{v u ' - u v '}{v} ^ 2.$

$\therefore f ' \left(x\right) = \frac{\left({e}^{3 x} - 5 x\right) \left({e}^{x}\right) ' - {e}^{x} \left({e}^{3 x} - 5 x\right) '}{{e}^{3 x} - 5 x} ^ 2 = \frac{\left({e}^{3 x} - 5 x\right) {e}^{x} - {e}^{x} \left(3 \cdot {e}^{3 x} - 5\right)}{{e}^{3 x} - 5 x} ^ 2 = \frac{{e}^{x} \left\{{e}^{3 x} - 5 x - 3 {e}^{3 x} + 5\right\}}{{e}^{3 x} - 5 x} ^ 2 = \frac{{e}^{x} \left(5 - 5 x - 2 {e}^{3 x}\right)}{{e}^{3 x} - 5 x} ^ 2.$